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40S Applied Math Mr. Knight – Killarney School Slide 2 Unit: Probability Lesson: PR-L1 Intro To Probability Some things that happen around us may be random events. That means that we have no way of knowing what will happen. For example, we may have no way of predicting whether a new baby will be a boy or a girl. Other events may be more predictable. For example, the likelihood of rolling the sum of 7 when rolling two dice is greater than the likelihood of rolling a sum of 10. The study of probability was started originally to analyze gambling problems. Today the same concepts are used to solve problems related to insurance, safety (e.g., helmet and seatbelt legislation), spread of disease (e.g., how many people will catch the flu), reliability of vehicles, and so on. Theory – Intro

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40S Applied Math Mr. Knight – Killarney School Slide 3 Unit: Probability Lesson: PR-L1 Intro To Probability Theory - Intro We sometimes hear statements like: "I will probably be home by 10:00 o'clock tonight," or "Today is not a good day for a picnic because it will likely rain," or "We are legally required to wear seatbelts because statistics show that we have a better chance of surviving a car crash if we wear our seatbelts." In each case, the statement is based on past experiences, which might include data from experiments. In this lesson, we will look at methods of determining both experimental and theoretical probabilities. We will also perform experiments by simulating situations using coins and spreadsheets.

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40S Applied Math Mr. Knight – Killarney School Slide 4 Unit: Probability Lesson: PR-L1 Intro To Probability When a teaspoon falls to the floor, it always comes to rest "right side up" or "right side down." In an experiment to determine how likely it is for the spoon to land "up" or "down," the spoon was dropped to the floor 40 times, and the results were as follows: What is the probability that the spoon comes to rest 'up'? Before we answer this question, we need to identify and define some words associated with probability. Theory – Definition of Probability

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40S Applied Math Mr. Knight – Killarney School Slide 5 Unit: Probability Lesson: PR-L1 Intro To Probability We say that the spoon experiment tested random outcomes because we had no way of knowing which way the spoon was going to land. The result of one trial (dropping the spoon once) is called an outcome. An event is a set of outcomes. For example, one event is the spoon landing "up." Probability may be defined as follows: Therefore, the probability of the spoon landing 'right side up' would be: We normally write probabilities as fractions reduced to lowest terms or in decimal form. Theory – Probability Formula

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40S Applied Math Mr. Knight – Killarney School Slide 6 Unit: Probability Lesson: PR-L1 Intro To Probability In the previous experiment where the spoon was dropped, the spoon never came to rest vertically, and it does not seem reasonable that it ever would do so. We say the spoon has no chance, or zero probability, of landing vertically. Therefore, The spoon always came to rest 'up' or 'down', and so we say that landing in one of these two ways was a certainty, and that the probability of landing in either one of these two ways was 100%. Therefore, All other probabilities must be expressed as values between zero and one. Therefore, Theory – Probability: Basic Properties

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40S Applied Math Mr. Knight – Killarney School Slide 7 Unit: Probability Lesson: PR-L1 Intro To Probability The probability that an event does not occur is called the complement of a probability, and is written as where E refers to the event. In this case, E refers to the spoon coming to rest in the 'up' position. Therefore, the probability that the spoon would not land 'up' is: Theory – Complements

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40S Applied Math Mr. Knight – Killarney School Slide 9 Unit: Probability Lesson: PR-L1 Intro To Probability Sometimes the calculation of probability is based on experimental data or on observations over a period of time. For example, the probability of birds migrating on a particular date can be based on the observed migration dates of previous years. Probabilities determined in this way are Experimental Probabilities. Do the following experiment (next slide). Drop a paper or foam cup to the floor 50 times, and record the number of times it comes to rest on its base, its side, and its top. A table similar to the one shown may be used to collect data. Determine the experimental probabilities of the cup landing on its side and on its top. Write your answers in decimal form rounded to two decimal places. Note: The probabilities calculated in this situation are experimental because the calculation of the probability of each event is based on experimental data. Theory – Experimental Probabilities

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40S Applied Math Mr. Knight – Killarney School Slide 11 Unit: Probability Lesson: PR-L1 Intro To Probability What is the probability that there will be two girls in a family with three children? To determine this experimentally, we can use three coins (three pennies) to represent the three children, with heads representing the girls. Toss the three coins and record the results. Two heads and one tail would represent a family with two girls and one boy. Repeat your coin tosses until you have 50 trials. A table like the one shown could be used to record the data. Determine the experimental probability of having two girls and one boy in a family with three children. Later in this module we will determine the theoretical probability of a family with three children having two girls and one boy. The theoretical probability for this kind of family is approximately 0.38. Compare your answer to the theoretical value. Do you think that your answer would improve if you used more trials? Theory – Experimental Probabilities

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40S Applied Math Mr. Knight – Killarney School Slide 12 Unit: Probability Lesson: PR-L1 Intro To Probability When we flip a coin, we can assume that there are equal chances of the coin landing heads or tails. Therefore, when flipping a coin, Flip a coin 10 times and count the number of times it lands 'Heads.' (It might be quicker to 'flip' 10 coins at one time by shaking them in a cup and dropping them onto a table.) Repeat this several times. Do the coins always land heads five times out of 10? Your results would probably be better if you flipped the coin 100 times. This, unfortunately, would take a lot of time and would be a very tedious experiment. For this reason, we use technology such as a graphing calculator or computer software to do the trials for us. We will use a spreadsheet program to simulate flipping 100 coins. Spreadsheet programs have the ability to generate random numbers. Theory – Using IT to Simulate Flipping Coins

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40S Applied Math Mr. Knight – Killarney School Slide 13 Unit: Probability Lesson: PR-L1 Intro To Probability In Excel, enter =Int(Rand()*2+1) into cell A1. Then copy the formula into cells B1 to J1 (you may use the "Fill Right" command), and then down 10 rows in columns A to J (using the "Fill Down" command). The formula will now randomly generate the numbers 1 and 2 in 100 cells. Your spreadsheet should look something like this: If we assign '1' to represent heads and '2' tails, we can find the experimental probability of flipping heads. Check your data to see whether you have approximately 50 'heads'. Theory – Using IT to Simulate Flipping Coins

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40S Applied Math Mr. Knight – Killarney School Slide 14 Unit: Probability Lesson: PR-L1 Intro To Probability The Cracker Corn Pops Company includes a prize in one out of every five boxes of Pops. We want to design an experiment to estimate the probability of getting at least two prizes when we buy six boxes of Pops for six people at a birthday party. We will simulate this experiment by using a spreadsheet program to generate random numbers from 1 to 5. The number 1 will represent a box with a prize. Create the following spreadsheet: Theory – Using IT to Simulate Flipping Coins

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40S Applied Math Mr. Knight – Killarney School Slide 15 Unit: Probability Lesson: PR-L1 Intro To Probability Enter =Int(Rand()*5+1) into cell A2. Then copy the formula into cells B2 to F2. Finally, to simulate 50 trials, copy the formulas from row 2 all the way to row 51. Each cell from A2 to F51 should now have a number from 1 to 5. Now complete column G by entering =(COUNTIF(A2:F2,1)) at G2 and filling down to row 51. To determine the probability of receiving at least two prizes in six boxes, count the number of times the number in column G is 2 or greater. Now do the following calculation: Theory – Using IT to Simulate Flipping Coins

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40S Applied Math Mr. Knight – Killarney School Slide 16 Unit: Probability Lesson: PR-L1 Intro To Probability A bag contains 12 marbles (5 red, 4 blue, and 3 green). What is the probability of randomly selecting a red marble if one marble is taken from the bag? Note: "randomly selecting" means that any marble in the bag has an equal chance of being selected. We can determine the theoretical probability by using the following formula: Theoretically we would select a red marble 5/12th (or about 42%) of the time if we repeated this activity many times. Theory – Theoretical Probabilities

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40S Applied Math Mr. Knight – Killarney School Slide 17 Unit: Probability Lesson: PR-L1 Intro To Probability The formula for theoretical probability could be written as follows: Note: This formula for theoretical probability works only if all outcomes are equally likely. For example, a die is equally likely to show any number from one to six. We cannot calculate the theoretical probability of the cup-dropping activity because the three outcomes are not equally likely. Theory – Theoretical Probability Formula